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 A354225 Lexicographically earliest sequence of distinct positive integers such that a(1) = 1 and for any n > 1, n / gcd(n, a(n)) and a(n) / gcd(n, a(n)) are prime. 1
 1, 3, 2, 6, 7, 4, 5, 12, 15, 14, 13, 8, 11, 10, 9, 24, 19, 27, 17, 28, 33, 26, 29, 16, 35, 22, 18, 20, 23, 42, 37, 48, 21, 38, 25, 54, 31, 34, 51, 56, 43, 30, 41, 52, 63, 58, 53, 32, 77, 70, 39, 44, 47, 36, 65, 40, 69, 46, 61, 84, 59, 74, 45, 96, 55, 78, 71 (list; graph; refs; listen; history; text; internal format)
 OFFSET 1,2 COMMENTS This sequence is a self-inverse permutation of the positive integers that preserves the number of prime divisors (with or without multiplicity). LINKS Michael De Vlieger, Table of n, a(n) for n = 1..10000 Michael De Vlieger, Annotated log-log plot of a(n), n = 1..2^14, showing records in red, local minima in blue, highlighting primes in green, fixed points in gold, and composite prime powers in magenta. Index entries for sequences that are permutations of the natural numbers FORMULA a(prime(2*n)) = prime(2*n-1) (where prime(n) denotes the n-th prime number). EXAMPLE The first terms are: n a(n) g=gcd(n, a(n)) n/g a(n)/g -- ---- -------------- --- ------ 1 1 1 1 1 2 3 1 2 3 3 2 1 3 2 4 6 2 2 3 5 7 1 5 7 6 4 2 3 2 7 5 1 7 5 8 12 4 2 3 9 15 3 3 5 10 14 2 5 7 11 13 1 11 13 12 8 4 3 2 13 11 1 13 11 14 10 2 7 5 MATHEMATICA nn = 120; c[_] = 0; a[1] = c[1] = 1; u = 2; Do[k = u; While[Nand[c[k] == 0, AllTrue[{i/#, k/#}, PrimeQ] &@ GCD[i, k]], k++]; Set[{a[i], c[k]}, {k, i}]; If[k == u, While[c[u] > 0, u++]], {i, 2, nn}]; Array[a, nn] (* Michael De Vlieger, May 22 2022 *) PROG (PARI) s=0; for (n=1, 67, for (v=1, oo, if (!bittest(s, v) && (n==1 || (isprime(n/g=gcd(n, v)) && isprime(v/g))), print1 (v", "); s+=2^v; break))) CROSSREFS Cf. A122280. Sequence in context: A154442 A375183 A371961 * A154445 A165199 A371962 Adjacent sequences: A354222 A354223 A354224 * A354226 A354227 A354228 KEYWORD nonn AUTHOR Rémy Sigrist, May 20 2022 STATUS approved

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Last modified September 18 13:18 EDT 2024. Contains 376000 sequences. (Running on oeis4.)